Theorem tgidm 14748

Description: The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
tgidm	⊢ (𝐵 ∈ 𝑉 → (topGen‘(topGen‘𝐵)) = (topGen‘𝐵))

Proof of Theorem tgidm

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgvalex 13296	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V)
2		eltg3 14731	. . . . 5 ⊢ ((topGen‘𝐵) ∈ V → (𝑥 ∈ (topGen‘(topGen‘𝐵)) ↔ ∃𝑦(𝑦 ⊆ (topGen‘𝐵) ∧ 𝑥 = ∪ 𝑦)))
3	1, 2	syl 14	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ (topGen‘(topGen‘𝐵)) ↔ ∃𝑦(𝑦 ⊆ (topGen‘𝐵) ∧ 𝑥 = ∪ 𝑦)))
4		uniiun 4019	. . . . . . . . . 10 ⊢ ∪ 𝑦 = ∪ 𝑧 ∈ 𝑦 𝑧
5		simpr 110	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → 𝑦 ⊆ (topGen‘𝐵))
6	5	sselda 3224	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ (topGen‘𝐵))
7		eltg4i 14729	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (topGen‘𝐵) → 𝑧 = ∪ (𝐵 ∩ 𝒫 𝑧))
8	6, 7	syl 14	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) ∧ 𝑧 ∈ 𝑦) → 𝑧 = ∪ (𝐵 ∩ 𝒫 𝑧))
9	8	iuneq2dv 3986	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → ∪ 𝑧 ∈ 𝑦 𝑧 = ∪ 𝑧 ∈ 𝑦 ∪ (𝐵 ∩ 𝒫 𝑧))
10	4, 9	eqtrid 2274	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → ∪ 𝑦 = ∪ 𝑧 ∈ 𝑦 ∪ (𝐵 ∩ 𝒫 𝑧))
11		iuncom4 3972	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ 𝑦 ∪ (𝐵 ∩ 𝒫 𝑧) = ∪ ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧)
12	10, 11	eqtrdi 2278	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → ∪ 𝑦 = ∪ ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧))
13		inss1 3424	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵
14	13	rgenw 2585	. . . . . . . . . . 11 ⊢ ∀𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵
15		iunss 4006	. . . . . . . . . . 11 ⊢ (∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵 ↔ ∀𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵)
16	14, 15	mpbir 146	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵
17	16	a1i 9	. . . . . . . . 9 ⊢ (𝑦 ⊆ (topGen‘𝐵) → ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵)
18		eltg3i 14730	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵) → ∪ ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ∈ (topGen‘𝐵))
19	17, 18	sylan2 286	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → ∪ ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ∈ (topGen‘𝐵))
20	12, 19	eqeltrd 2306	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → ∪ 𝑦 ∈ (topGen‘𝐵))
21		eleq1 2292	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 ∈ (topGen‘𝐵) ↔ ∪ 𝑦 ∈ (topGen‘𝐵)))
22	20, 21	syl5ibrcom 157	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → (𝑥 = ∪ 𝑦 → 𝑥 ∈ (topGen‘𝐵)))
23	22	expimpd 363	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → ((𝑦 ⊆ (topGen‘𝐵) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ (topGen‘𝐵)))
24	23	exlimdv 1865	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦(𝑦 ⊆ (topGen‘𝐵) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ (topGen‘𝐵)))
25	3, 24	sylbid 150	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ (topGen‘(topGen‘𝐵)) → 𝑥 ∈ (topGen‘𝐵)))
26	25	ssrdv 3230	. 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘(topGen‘𝐵)) ⊆ (topGen‘𝐵))
27		bastg 14735	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵))
28		tgss 14737	. . 3 ⊢ (((topGen‘𝐵) ∈ V ∧ 𝐵 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) ⊆ (topGen‘(topGen‘𝐵)))
29	1, 27, 28	syl2anc 411	. 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ⊆ (topGen‘(topGen‘𝐵)))
30	26, 29	eqssd 3241	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘(topGen‘𝐵)) = (topGen‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  Vcvv 2799   ∩ cin 3196   ⊆ wss 3197  𝒫 cpw 3649  ∪ cuni 3888  ∪ ciun 3965  ‘cfv 5318  topGenctg 13287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-topgen 13293

This theorem is referenced by:  tgss3  14752  txbasval  14941